let

a = the days that 1 adult needs to finish a doghouse

1/a house per days is the rate of the adult

k = the days that 1 kid needs to finish a doghouse

1/k house per days is the rate of the kd

2 adult and 3 kids can build a doghouse in 16 days

2/a + 3/k = 1/16

3 adults and 7 kids finish it in 10 days

3/a + 7/k = 1/10

multiply the first equation by 3 and subtract the second equation multiplied by 2 and you will get

3(2/a + 3/k) - 2(3/a + 7/k) = 3/16 - 2/10

6/a + 9/k - 6/a - 14/k = 3/16 - 1/5

9/k - 14/k = 3/16 - 1/5

by solving the equation we find

k = 400 days

click here to see the step by step solution of the equation

one kid will finish the house in 400 days thus 20 kids will finish the dog house 20 times faster in 20 times less days

400/20 = 20 days

20 kids will finish the house in 20 days.